docs: session headers name the harness and the model (displayModel)

- api-reference: the new `displayModel` session field, its sources in order
  (custom-endpoint, statusline, screen, launch) and that it is untrusted
  display text, persisted and restored when the CLI reported it.
- cli-registry: `capabilities.modelDetect` (one capture group, the last rows
  of the probe's capture, anchored on chrome only that CLI draws), the two
  stock patterns (dsh-TUI, codex) and the fifth config regex.
- architecture-invariants (tile grid): the header painter, the id as data, the
  untrusted model text, no writes for an unchanged session, the truncation
  order, Pane A's strip and its fits through syncTerminalGeometry.
- tile-grid-plan "as built", the wiki's Tile Grid page (logo and model rows,
  Split's strips), and CLAUDE.md's tile grid and CLI registry paragraphs.

Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com>

docs/api-reference.md

@@ -484,6 +484,30 @@ also pure decoration: it confers no permission, and a child is unaffected by its
 parent exiting. It appears on session state as `parentSessionId` (absent when
 unresolved) and survives a server restart.
 
+## Session model (`displayModel`)
+
+Session state (`GET /api/v1/sessions`, the `session:updated` event) carries the model a
+session runs as far as the server knows it, for the web UI's session headers:
+
+```json
+"displayModel": { "model": "qwen3.8-27b", "source": "screen" }
+```
+
+`source` is where it came from, strongest first:
+
+| `source`          | Meaning                                                                                               |
+| ----------------- | ----------------------------------------------------------------------------------------------------- |
+| `custom-endpoint` | The session is pointed at a Custom Model Endpoint Profile; its `modelId` answers, whatever the CLI prints. |
+| `statusline`      | Claude's statusLine exporter reported it (`model.display_name`); follows an in-session `/model`.          |
+| `screen`          | Read off the CLI's own footer (`capabilities.modelDetect`, today dsh and codex); follows a switch.      |
+| `launch`          | What the session was launched with (`--model`, the app-wide default, `<cli>Config.model`); nothing has reported since. |
+
+Between `statusline` and `screen` the newest report wins. The field is absent when no
+model is known (a shell, a CLI that reports none and was launched without one). `model`
+is display text from a pane or a CLI report: control characters are stripped and it is at
+most 64 characters, but treat it as untrusted text. A `statusline` or `screen` value is
+persisted and restored after a server restart until the next report replaces it.
+
 ## Approvals Inbox
 
 Cross-session queue of prompts waiting on a human (permission dialogs,